We derive robust constraints on primordial non-Gaussianity (PNG) using the clustering of 800 000 photometric quasars from the Sloan Digital Sky Survey in the redshift range 0.5<z<3.5. These measurements rely on the novel technique of extended mode projection to control the impact of spatially varying systematics in a robust fashion, making use of blind analysis techniques. This allows the accurate measurement of quasar halo bias at the largest scales, while discarding as little as possible of the data. The standard local-type PNG parameters f_{NL} and g_{NL} both imprint a k^{-2} scale-dependent effect in the bias. Constraining these individually, we obtain -49<f_{NL}<31 and -2.7×10^{5}<g_{NL}<1.9×10^{5}, while their joint constraints lead to -105<f_{NL}<72 and -4.0×10^{5}<g_{NL}<4.9×10^{5} (all at 95% C.L.). Introducing a running parameter n_{f_{NL}} to constrain b(k)∝k^{-2+n_{f_{NL}}} and a generalized PNG amplitude f[over ˜]_{NL}, we obtain -45.5 exp(3.7n_{f_{NL}})<f[over ˜]_{NL}<34.4 exp(3.3n_{f_{NL}}) at 95% C.L. These results incorporate uncertainties in the cosmological parameters, redshift distributions, shot noise, and the bias prescription used to relate the quasar clustering to the underlying dark matter. These are the strongest constraints obtained to date on PNG using a single population of large-scale structure tracers, and are already at the level of pre-Planck constraints from the cosmic microwave background. A conservative forecast for a Large Synoptic Survey Telescope (LSST)-like survey incorporating mode projection yields σ(f_{NL})∼5-competitive with the Planck result-highlighting the power of upcoming large scale structure surveys to probe the initial conditions of the Universe.